Elliptic curves of unit discriminant over real quadratic number fields


Table of curves with irrational j
More about isogenies and torsion
The height proxy H and the search
Curves with rational j, and the m=6 table
References

Table of curves with irrational j

This file lists 621 elliptic curves E/F where: Suppose E is given by the extended Weierstrass equation
y2 + a1 x y + a3 y = x3 + a2 x2 + a4 x + a6
of discriminant Δ, with coefficients ai in the ring of integers OF. Then for every unit u the equation
y2 + u a1 x y + u3 a3 y = x3 + u2 a2 x2 + u4 a4 x + u6 a6
is an equation for an isomorphic curve with discriminant u12Δ. Hence we may assume that Δ = ±ηm where η is a fundamental unit and 0 ≤ m < 12. The argument of [Stroeker 1983] shows (though the search did not assume this) that m=0 cannot occur. [I do not know whether this is noted in the literature. The main result of [Stroeker 1983] is that there is no elliptic curve of unit discriminant with a global minimal model over an imaginary quadratic field; but all that is used about such fields is that the discriminant would have to be ±1 (except for the two cyclotomic fields, for which an additional argument is given), and then the same analysis yields the impossibility of a Δ=±1 curve also in the real quadratic case.] It seems that m=6 is also impossible unless j is rational; perhaps this is proved as well. At any rate I found only examples with m in {±1, ±2, ±3, ±4, ±5}. If E satisfies our conditions on D, Δ, and j, then so does the Galois conjugate of E. If E has discriminant ±ηm then the Galois conjugate has discriminant ±η−m (not necessarily with the same choice of sign). Hence we need only list curves with m in {1, 2, 3, 4, 5}, and Galois conjugation will double the roster.

I list each curve with some additional information in the format

[D, σ, m, s, j, H, [a1, a2, a3, a4, a6], "q", d],
in which the elements j=j(E) and ai of F are written in terms of the standard generator w of OF over Z (so  w2 = D/4  or  w2 = w + (D−1)/4  according as D is even or odd), and the other entries are as follows: I also tested for isogenies of degree 13, but found none (and even 7-isogenies arose only for discriminant D=28 — more on this below). There might be isogenies of other degrees, but there are no further Q-curves, except those for D=33, 41, and 65 noted in the next section: for each of the remaining curves one readily finds a rational prime p split in F such that the traces of E modulo the two primes above p are neither equal nor each other's negative; this proves that E is not isogenous with any quadratic twist of its Galois conjugate.

More about isogenies and torsion

There are only 28 nontrivial isogeny classes in all, and for 22 of them there is only one isogeny:
  • l=2: eight Q-curves, for D = 337, 881, 2657, 6817, 14897, 28817, 50881, 130577; and four 2-isogenies not involving Q-curves, for discriminants D = 4092, 4092, 93193, 93193, with the repetitions at D=4092 and D=93193 due to quadratic twists by the square roots of −1 and η respectively. [In general twists can be detected by coincidence of j values (up to Galois conjugation, because the table lists only one of each pair of Galois-conjugate curves), and the H entries match as well.] Curves with a 2-torsion point and a global minimal model of unit discriminant over a quadratic number field are the topic of [Comalada 1990]; the D=4092 curves are the special case d=1023 of Theorem 1(iii) in that paper.
  • l=3: two Q-curves, for D=109 and D=997, and three 3-isogenies not involving Q-curves, for discriminants D = 733, 18541, and 193189; and
  • l=5: four Q-curves, for D = 349, 461, 509, and 1709, and one 5-isogeny not involving Q-curves, also for discriminant D=509.

    • The remaining cases are those of the six smallest discriminants appearing in the table, D=24, 28, 29, 33, 41, and 65. The curves for D=24, 28, 33 have complex multiplication (CM) and are 2- or 3-isogenous with CM curves of rational j-invariant; these are the only CM curves in the table. The curves for D=29, 41, and 65 are non-CM Q-curves. We next describe these isogeny classes in more detail. First the CM curves:

    i) For D=24 there are two CM curves each with endomorphism ring Z[3√−2]. […]

    ii) For D=28 there are two CM curves, each with endomorphism ring Z[2√−7]. They are quadratic twists by sqrt(−1). Since here F contains sqrt(7), this makes them also quadratic twists by sqrt(−7), which generates the CM field, so these curves are related by a 28-isogeny. In particular, they admit 7-isogenies; this, too, is unique in the table. Each curve also has a 2-isogeny, so d=14 for these curves. The 7-isogeny from each curve goes to its Galois conjugate, as indicated by the "Qcurve_7" tag. The 2-isogenous curves are also not tabulated because they are CM curves with endomorphism ring Z[√−7], and thus rational j-invariant, namely 2553. These j=2553 curves are again Galois conjugate and 7-isogenous; each also has all of its Each of these has all of its 2-torsion rational over F, and thus has three 2-isogenies to curves with everwhere good reduction over F: one of the curves in the table, its Galois conjugate, and a third curve with j-invariant −153 and endomorphisms by the full ring of integers in Q(√−7).

    iii) For D=33 there are two CM curves each with endomorphism ring Z[3(1+√−11)/2]. As in (i), they are 3-isogenous to CM curves with rational j-invariant, here −323. […]

    Next the non-CM curves with more than one isogeny:

    i) For D=29 there are two isogenous pairs of Q-curves, each related to its Galois conjugate by a 5-isogeny and to the other pair by 3-isogenies; therefore d=15 for these curves.

    ii) For D=41 there is a Q-curve with all its 2-torsion points rational over F; thus it has three isogenies of degree 2, so d=8. One of these isogenies goes to the Galois conjugate, and the others go to two other curves each 8-isogenous with its Galois conjugate. (These curves are thus also Q-curves, though the q string doesn't record this because I didn't have GP check directly for an 8-isogeny to the Galois conjugate.)

    iii) Finally, each of the D=65 curves is 2-isogenous with a curve which is not tabulated because its j-invariant is rational. Thus the D=65 curves in the table are Q-curves and again their q strings don't record this fact because I didn't have GP check directly for an isogeny of composite degree (degree 4 in this case). The 2-isogenous curves are [1+w, w, 1+w, 5+2*w, −23−6*w] (with j = 4913 = 173) and [1, −1+w, 1+w, −600−172*w, −9422−2667*w] (with j = 2573), and they are 2-isogenous with each other. Thus we have in total six isogenous curves of unit conductor over F, related by 2-isogenies forming a graph of shape ⟩−⟨ . This is the same shape as for the D=41 curves, but with a different Galois action (since none of the D=41 curves has j rational).

    It is shown in [Comalada 1990, Theorem 2] that if an elliptic curve has a global minimal model of unit discriminant over a quadratic number field, and has all its 2-torsion points rational over that field, then it is one of the six curves with this property noted for D=28, 41, and 65 (the d=8 curve for D=41, and the curves with j-invariant 2553 for D=28 and 173, 2573 for D=65). [That theorem also determines all curves with full 2-torsion over a quadratic field that have unit discriminant even if there is no global minimal model; this adds only two more examples, the twists of the two D=65 curves by sqrt(5).]

    In several cases the table contains a curve E with an isogeny to a curve E', but E' does not appear in the table. Usually this is because E' has discriminant ±ηm with m in {−1, −2, −3, −4, −5}, and thus the table contains not E' but its Galois conjugate. For example this happens whenever E is a Q-curve and E' is its Galois conjugate. Rarer and more interesting is what happens for the 2-isogenies for D=93193 and the 3-isogenies for D=18541 and 193189: the isogenous curve, though of unit conductor, does not have a global minimal model. That is, there is no Weierstrass equation with coefficients in OF and unit discriminant.

    For a general elliptic curve E over a number field F, it is known that there is an obstruction δ, contained in the 12-torsion of the class group of F, to the existence of a Weierstrass equation with coefficients in OF that is minimal at each prime. At best one can choose an ideal D in the class of δ such that E has good reduction at each prime in D, and find an equation for E whose discriminant generates the ideal D12Δ(E) but has coefficients ai in Di for each i = 1, 2, 3, 4, 6. Such E are easy to construct, and [Cremona 1992] already contains examples where E has good reduction everywhere over a real quadratic field but δ is a nontrivial 3-torsion element of the class group (for the Q-curves over the fields of discriminant 229 and 257). The new phenomenon here is that δ is not invariant under isogeny, even for isogenies from curves E/F with unit discriminant. This can happen when the degree l of the isogeny splits in F and the kernel of isogeny reduces to the identity modulo one of the primes above l but not the other. If these primes are not principal then the isogenous curve has nontrivial δ.

    Torsion. Whenever two curves over F are related by isogenies of degree 2, the kernel of the isogeny contains an F-rational 2-torsion point. Besides those, I found only three more pairs of nontrivial torsion points:

    If the list of isogenies is complete then so is the list of torsion points.

    The height proxy H and the search

    Given m (and thus |Δ|), the formula H = D−1/2 ((c4|Δ|−1/3)2 + (ιc4|ιΔ|−1/3)2) makes H, considered as a function of c4, a positive-definite quadratic form on OF. Thanks to the normalizing factor D−1/2, this form has discriminant 1. If H ≤ N then c43 ± 1728Δ has norm at most (D1/4(N/2)1/2+1728)6. Thus, for each of the roughly πN elements c4 of OF in the ellipse H≤N, we might expect to find a solution of c43 − c62 = 123Δ with probability proportional to D−3/4H−3/2, and the number of solution we expect to miss is of size D−3/4N−1/2 for any individual D, and so D1/4N−1/2 when summed over all discriminants up to D. A positive fraction of those solutions (depending on the classes of c4 and c6 mod some powers of 2 and 3) will then give rise to curves of unit discriminant. For each D, our algorithm thus finds a reduced basis for the quadratic form H, and tries all candidates up to N. I took N about 15000, extending the search to 120000 for D < 105. (Actually I tried all candidates in a parallelogram with sides parallel to the reduced lattice basis and long enough [H=4N/3 in each direction] to guarantee that it contains the ellipse; this also caught a few curves with H>N, and naturally I retained those extra curves.) The extended search added only 22 curves to the 240 found with the lower threshold: the Q-curve for D=349, with l=5, and curves without known nontrivial isogenies for D=9704, 13233*, 14261, 14984*, 17261, 21665*, 23861*, 24265*, 29105*, 30764*, 48545*, 50433* (the “*” marks a pair of curves related by quadratic twist; NB for D=17261 there are two curves but no “*” because these curves are not twists, and indeed one of them was easily found with the lower threshold because it has H<35).

    This still leaves 14 of the 621 tabulated curves that were beyond the search limit. Four of them, the CM curves for D=24 and D=28, come from [Colamada 1990], which suggested the two CM curves with D=33; two more, the Q-curves for D=509 and D=997 (isogeny degree 5 and 3 respectively), come from [Cremona 1992]. The other six are isogenous to curves within the search range: one curve in each of the isogeny classes of Q-curves for D=29, 41, and 65; one of the two curves involved in the 3-isogeny for D=733; and one curve in each of the 2-isogenies for D=4092. The entry for each of these fourteen curves is indented two spaces in the table.

    I wouldn't be surprised if there are on the order of 10-20 further curves, including some Q-curves, that belong in the table but were beyond the search range and could not be supplied using these other methods. [Added a few days later: [Umegaki 1998] gives such a construction and gives several examples, including one with a 7-isogeny for D=497. See Table 1 on page 197 for a list of discriminants up to 1000. I'll add those in due course. Umegaki also reports (p.192, footnote) that R.Pinch proved in his Ph.D. thesis (Oxford, 1982) that there are no 13-isogenous Q-curves of unit discriminant over a quadratic number field.]

    Curves with rational j, and the m=6 table

    Besides the curves in the table, the search also turned up many more curves with rational j-invariant — to the point that the ones with irrational j are only about 2% of the total. When Δ is a unit (or even a 12th power, if the obstruction δ is nontrivial), a rational j must be a cube: Δ·j is a cube in K (namely the cube of c4/12); taking norms we deduce that j2, and thus also j, is a cube in Q. Thus in turn m must be a multiple of 3. In almost all cases m = ±3; the other possible m value of 6 arose only 91 times (listed     here, in the same format without the final two entries). Curves with rational j that have everywhere good reduction over a quadratic number field (whether imaginary or real, and whether or not δ is trivial) were studied in some detail in [Setzer 1981], whose Theorem 1 already contains the observation that j must be a cube.

    Almost 90% of the curves with rational j are accounted for by six CM values of the j-invariant: most commonly −153 and 2553 (CM discriminants −7 and −28), followed by 203, −323, −963, and −9603 (CM discriminants −8, −11, −19, and −43 respectively; the search region was not large enough to catch curves of CM discriminant −67 and −163). These were followed by the non-CM invariants j=393, 163, −163, 173, 2573, and −133, with multiplicities ranging from 797 down to 128, and many others, for a total of 99 values of j, half of which arose no more than three times (29 once, 12 twice, and 9 thrice).

    The CM invariants j = −153 (discriminant −7) and j = 2553 (discriminant −28) also appear twice each in the m=6 list, accounting for the four curves over the field Q(71/2) of discriminant +28. For each of these two values of j, the two curves are quadratic twists of each other by the square root of −1.

    There are 20 more examples in the m=6 list where more than one curve appears for the same D. In all but one of these cases, namely those with D = 316, 5980, 7516, 8284, 41756, 55580, 63964, 71164, 112252, 126076, 167356, 252316, 291004, 374908, 380060, 408380, 413116, 490076, and 672412, there are two curves related by a sqrt(−1) twist. In the remaining case, with D = 65, we find two curves with j = 173 and 2573 related by a 2-isogeny. Testing all the curves in the table for isoegnies of degree 2, 3, 5, 7, or 13 found only two more examples: D=37, with j-invariant 4096, and D=104, with j-invariant 64. In each case there is an isogeny of degree 5 to a curve that lies well beyond the search range for such a small D; the isogenous curves have j-invariants 33763 and −28763 respectively. The D=37 curves are already in [Comalada 1990], together with the observation that the curve with j = 4096 has a 5-torsion point in the kernel of the isogeny. The j = 64 curve for D=104 has the same property.

    References

    [Comalada 1990] Comalada, S.:     Elliptic curves with trivial conductor over quadratic fields, Pacific J. Math. 144 (1990) #2, 237–258.
    [Cremona 1992] Cremona, J.E.: Modular symbols for Γ1(N) and elliptic curves with everywhere good reduction, Math. Proc. Cambridge Philos. Soc. 111 (1992) #2, 199–218.
    [Kagawa 1998] Kagawa, T.: Determination of elliptic curves with everywhere good reduction over Q(√37), Acta Arithmetica 83 (1998) #3, 253–269.
    [Setzer 1981] Setzer, B.: Elliptic curves with good reduction everywhere over quadratic fields and having rational j-invariant, Illinois J. Math. 25 (1981) #2, 233–245.
    [Stroeker 1983] Stroeker, R.J.: Reduction of elliptic curves over imaginary quadratic number fields, Pacific J. Math. 108 (1983) #3, 451–463.
    [Umegaki 1998] Umegaki, A.: A Construction of Everywhere Good Q-Curves with p-Isogeny, Tokyo J. Math. 21 (1998) #1, 183–200.